Problem: Simplify and expand the following expression: $ \dfrac{4}{3a - 27}- \dfrac{4}{2a - 8}+ \dfrac{5a}{a^2 - 13a + 36} $
Explanation: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $3$ out of denominator in the first term: $ \dfrac{4}{3a - 27} = \dfrac{4}{3(a - 9)}$ We can factor a $2$ out of denominator in the second term: $ \dfrac{4}{2a - 8} = \dfrac{4}{2(a - 4)}$ We can factor the quadratic in the third term: $ \dfrac{5a}{a^2 - 13a + 36} = \dfrac{5a}{(a - 9)(a - 4)}$ Now we have: $ \dfrac{4}{3(a - 9)}- \dfrac{4}{2(a - 4)}+ \dfrac{5a}{(a - 9)(a - 4)} $ The least common multiple of the denominators is: $ 6(a - 9)(a - 4)$ In order to get the first term over $6(a - 9)(a - 4)$ , multiply by $\dfrac{2(a - 4)}{2(a - 4)}$ $ \dfrac{4}{3(a - 9)} \times \dfrac{2(a - 4)}{2(a - 4)} = \dfrac{8(a - 4)}{6(a - 9)(a - 4)} $ In order to get the second term over $6(a - 9)(a - 4)$ , multiply by $\dfrac{3(a - 9)}{3(a - 9)}$ $ \dfrac{4}{2(a - 4)} \times \dfrac{3(a - 9)}{3(a - 9)} = \dfrac{12(a - 9)}{6(a - 9)(a - 4)} $ In order to get the third term over $6(a - 9)(a - 4)$ , multiply by $\dfrac{6}{6}$ $ \dfrac{5a}{(a - 9)(a - 4)} \times \dfrac{6}{6} = \dfrac{30a}{6(a - 9)(a - 4)} $ Now we have: $ \dfrac{8(a - 4)}{6(a - 9)(a - 4)} - \dfrac{12(a - 9)}{6(a - 9)(a - 4)} + \dfrac{30a}{6(a - 9)(a - 4)} $ $ = \dfrac{ 8(a - 4) - 12(a - 9) + 30a} {6(a - 9)(a - 4)} $ Expand: $ = \dfrac{8a - 32 - 12a + 108 + 30a}{6a^2 - 78a + 216} $ $ = \dfrac{26a + 76}{6a^2 - 78a + 216}$ Simplify: $ = \dfrac{13a + 38}{3a^2 - 39a + 108}$